How many of the following pairs of function are identcal
(a) f(x)=x2x, g(x)=x
(b) f(x)=x2−1x−1, g(x)=x+1
(c) f(x)=sec2x−tan2x, g(x)=1
(d) f(x)=xx2, g(x)=1x
Two functions are equal and identical if the domains and range are equal.
[f(x) = g(x) for every x in their domain]
1. Domain of f (x) is real numbers without zero or R - {0} because x2x is not defined as zero.
f (x) and g(x) are not equal because their domains are not equal.
2. For this pair also the domains are not the same because x2−1x−1 is not defined at x = 1.
3. Domain of g(x) is R (real numbers) f(x) is not defined when x = (2n + 1)π2.
⇒ they are not equal
4. xx2 is not defined at x = 0.
If x ≠ 0, xx2=1x
1x is not defined at x = 0
⇒ f (x) = g(x) = 1x when x≠0 . Since zero is not in their domain, the functions are identical.